To find the $t^*$ multiplier for a 99% confidence interval when the sample size is 45, we follow these steps:

1. Determine the degrees of freedom (df): The degrees of freedom for the $t$-distribution is given by $df = n - 1$, where $n$ is the sample size. $df = 45 - 1 = 44$

2. Locate the 99% confidence level in the $t$-distribution table: For a 99% confidence interval, the level of significance $\alpha$ is 0.01. Since the confidence interval is two-tailed, we split the significance level in half for each tail: $\alpha/2 = 0.01/2 = 0.005$

3. Find the $t$-value corresponding to $df = 44$ and $\alpha/2 = 0.005$: Using the $t$-distribution table or a calculator that provides $t$-values, we look up the $t$-value for $df = 44$ and $\alpha/2 = 0.005$.

Using a standard $t$-table or a statistical calculator, we find:

$t^* \approx 2.6955$

Thus, the $t^*$ multiplier for a 99% confidence interval when the sample size is 45 is approximately $2.6955$.

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. How is the degrees of freedom calculated in a $t$-distribution?
2. What is the difference between a $t$-distribution and a normal distribution?
3. How does the sample size affect the $t$-distribution?
4. What is the significance level $\alpha$ in hypothesis testing?
5. How do you interpret a 99% confidence interval in practical terms?
6. How would the $t^*$ value change if the sample size were increased to 100?
7. Why is the $t$-distribution used instead of the normal distribution for small sample sizes?
8. How do you use the $t$-distribution table to find critical values?

Tip: When using the $t$-distribution, always make sure to check the degrees of freedom and the corresponding $\alpha$ value to find the correct $t$-multiplier.